Page 321 - Schaum's Outline of Theory and Problems of Signals and Systems
P. 321
FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS AND SYSTEMS [CHAP. 6
Solved Problems
DISCRETE FOURIER SERIES
6.1. We call a set of sequences (qk[n]) orthogonal on an interval [N,, N,] if any two signals
W,[n] and qk[n] in the set satisfy the condition
where * denotes the complex conjugate and a # 0. Show that the set of complex
exponential sequences
is orthogonal on any interval of length N.
From Eq. (1.90) we note that
Applying Eq. (6.116), with a = eik(2"/N), we obtain
since e'k(2"/N'N = e jk2" = 1. Since each of the complex exponentials in the summation in
Eq. (6.117) is periodic with period N, Eq. (6.117) remains valid with a summation carried over
any interval of length N. That is,
k =O,f N,f 2N, ...
C eiWn/N)n,
otherwise
n=(N)
Now, using Eq. (6.118), we have
where m, k < N. Equation (6.119) shows that the set (e~~(~"/~)": k = 0.1,. . . , N - 1) is orthog-
onal over any interval of length N. Equation (6.114) is the discrete-time counterpart of
Eq. (5.95) introduced in Prob. 5.1.
